The Third British Mathematical Olympiad
1967年第三届英国奥林匹克数学竞赛 
 a, b are the roots of x^{2} + Ax + 1 = 0, and c, d are the roots of x^{2} + Bx + 1 = 0. Prove that (a  c)(b  c)(a + d)(b + d) = B^{2}  A^{2}.
 Graph x^{8} + xy + y^{8} = 0, showing stationary values and behaviour for large values. [Hint: put z = y/x.]
 (a) The triangle ABC has altitudes AP, BQ, CR and AB > BC. Prove that AB + CR ≥ BC + AP. When do we have equality ?
(b) Prove that if the inscribed and circumscribed circles have the same centre, then the triangle is equilateral.
 We are given two distinct points A, B and a line l in the plane. Can we find points (in the plane) equidistant from A, B and l? How do we construct them ?
 Show that (x  sin x)(π  x  sin x) is increasing in the interval (0, π/2).
 Find all x in [0, 2π] for which 2 cos x ≤ √(1 + sin 2x)  √(1  sin 2x) ≤ √2.
 Find all reals a, b, c, d such that abc + d = bcd + a = cda + b = dab + c = 2.
 For which positive integers n does 61 divide 5^{n}  4^{n }?
 None of the angles in the triangle ABC are zero. Find the greatest and least values of cos^{2}A + cos^{2}B + cos^{2}C and the values of A, B, C for which they occur.
 A collects pre1900 British stamps and foreign stamps. B collects post1900 British stamps and foreign special issues. C collects pre1900 foreign stamps and British special issues. D collects post1900 foreign stamps and British special issues. What stamps are collected by (1) no one, (2) everyone, (3) A and D, but not B ?
 The streets for a rectangular grid. B is h blocks north and k blocks east of A. How many shortest paths are there from A to B ?

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